# dbd: Probability Density Under Birth-Death Models In ape: Analyses of Phylogenetics and Evolution

## Description

These functions compute the probability density under some birth–death models, that is the probability of obtaining x species after a time t giving how speciation and extinction probabilities vary through time (these may be constant, or even equal to zero for extinction).

## Usage

 ```1 2 3 4``` ```dyule(x, lambda = 0.1, t = 1, log = FALSE) dbd(x, lambda, mu, t, conditional = FALSE, log = FALSE) dbdTime(x, birth, death, t, conditional = FALSE, BIRTH = NULL, DEATH = NULL, fast = FALSE) ```

## Arguments

 `x` a numeric vector of species numbers (see Details). `lambda` a numerical value giving the probability of speciation; can be a vector with several values for `dyule`. `mu` id. for extinction. `t` id. for the time(s). `log` a logical value specifying whether the probabilities should be returned log-transformed; the default is `FALSE`. `conditional` a logical specifying whether the probabilities should be computed conditional under the assumption of no extinction after time `t`. `birth, death` a (vectorized) function specifying how the speciation or extinction probability changes through time (see `yule.time` and below). `BIRTH, DEATH` a (vectorized) function giving the primitive of `birth` or `death`. `fast` a logical value specifying whether to use faster integration (see `bd.time`).

## Details

These three functions compute the probabilities to observe `x` species starting from a single one after time `t` (assumed to be continuous). The first function is a short-cut for the second one with `mu = 0` and with default values for the two other arguments. `dbdTime` is for time-varying `lambda` and `mu` specified as R functions.

`dyule` is vectorized simultaneously on its three arguments `x`, `lambda`, and `t`, according to R's rules of recycling arguments. `dbd` is vectorized simultaneously `x` and `t` (to make likelihood calculations easy), and `dbdTime` is vectorized only on `x`; the other arguments are eventually shortened with a warning if necessary.

The returned value is, logically, zero for values of `x` out of range, i.e., negative or zero for `dyule` or if ```conditional = TRUE```. However, it is not checked if the values of `x` are positive non-integers and the probabilities are computed and returned.

The details on the form of the arguments `birth`, `death`, `BIRTH`, `DEATH`, and `fast` can be found in the links below.

## Value

a numeric vector.

## Note

If you use these functions to calculate a likelihood function, it is strongly recommended to compute the log-likelihood with, for instance in the case of a Yule process, `sum(dyule( , log = TRUE))` (see examples).

## References

Kendall, D. G. (1948) On the generalized “birth-and-death” process. Annals of Mathematical Statistics, 19, 1–15.

`bd.time`, `yule.time`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28``` ```x <- 0:10 plot(x, dyule(x), type = "h", main = "Density of the Yule process") text(7, 0.85, expression(list(lambda == 0.1, t == 1))) y <- dbd(x, 0.1, 0.05, 10) z <- dbd(x, 0.1, 0.05, 10, conditional = TRUE) d <- rbind(y, z) colnames(d) <- x barplot(d, beside = TRUE, ylab = "Density", xlab = "Number of species", legend = c("unconditional", "conditional on\nno extinction"), args.legend = list(bty = "n")) title("Density of the birth-death process") text(17, 0.4, expression(list(lambda == 0.1, mu == 0.05, t == 10))) ## Not run: ### generate 1000 values from a Yule process with lambda = 0.05 x <- replicate(1e3, Ntip(rlineage(0.05, 0))) ### the correct way to calculate the log-likelihood...: sum(dyule(x, 0.05, 50, log = TRUE)) ### ... and the wrong way: log(prod(dyule(x, 0.05, 50))) ### a third, less preferred, way: sum(log(dyule(x, 0.05, 50))) ## End(Not run) ```